Distinguish a best response to one opponent choice from dominance across all choices.
By the end: test strict and weak dominance and recognize what deletion assumes.
Hold the opponent fixed
For Row in the maintenance game, compare 5 with 3 when Column maintains, and 1 with 0 when Column skips. Skip is strictly better in both comparisons. It strictly dominates Maintain. Column has the same incentive. No probability forecast about the other person is needed for these comparisons.
A best response maximizes your payoff against a specified opponent action or probability distribution. A dominant action is a best response against every opponent action. An action can be a best response in one column and poor in another. That is why selecting the largest number anywhere in the matrix is not a solution.
Compare differences within each opponent choice
A dominance test compares two actions belonging to the same player. It does not compare Row's payoff with Column's payoff inside one cell. For Row, keep one column fixed and subtract Maintain's payoff from Skip's. Against Maintain, the difference is five minus three, or two. Against Skip, it is one minus zero, or one. Both differences are positive, so Skip strictly dominates Maintain. The size of the gains differs, but their positive signs establish the claim. One positive difference and one negative difference would not establish dominance, however large either number was.
For Column, keep a row fixed and read the second coordinates. When Row maintains, Column gets five by skipping and three by maintaining, a gain of two. When Row skips, Column gets one by skipping and zero by maintaining, a gain of one. Symmetry makes these calculations match but is not required. A game can give only one player a dominant action. Always perform that player's own comparisons instead of assuming that an argument about Row automatically applies to Column too.
Strict dominance needs a positive gain in every comparison
Column fixed
Row Skip − Maintain
Maintain
5 − 3 = +2
Skip
1 − 0 = +1
Row fixed
Column Skip − Maintain
Maintain
5 − 3 = +2
Skip
1 − 0 = +1
Each subtraction uses one player's utility and holds the other action fixed. Both players have positive maintenance-game gains.
Best response has a smaller scope. In Joint project, Join gives Row six against Join while Safe gives four. Join is the best response to that particular action. Against Safe, Join gives zero while Safe gives four, so Safe is best instead. The payoff differences for Join minus Safe are positive two and negative four. Neither action dominates the other. Saying that six is the largest entry misses the choice problem because Row cannot force Column to Join. Dominance demands a comparison under each opponent action, not a comparison of the most attractive cells.
An action with the best worst outcome need not dominate either. The worst payoff from Safe is four, while the worst payoff from Join is zero. A rule that chooses the largest guaranteed minimum therefore chooses Safe. This rule is called maximin, because it maximizes the minimum payoff. Yet Join is better when Column joins. Maximin and dominance happen to agree on Skip in maintenance but disagree about what can be concluded in coordination. Selecting an action for its guarantee is an additional decision rule, not evidence that it wins every fixed-opponent comparison.
Beliefs about uncertain actions can help when dominance fails. Expected utility means multiplying each possible utility by its probability and adding the products. Here we assume the stipulated numbers represent preferences over uncertain outcomes, not merely ranks of certain outcomes. A probability of seventy-five percent means three quarters of the total weight. In the project game, that belief about Column joining gives Join expected utility six times three quarters plus zero times one quarter, or four and a half. Safe still gives four. This makes Join a best response to that belief, although Safe remains better against a certain Safe. Averaging has not turned a conditional advantage into dominance.
A guarantee and a probability-weighted comparison answer different questions
Row action
Column Join
Column Safe
Minimum
75% Join belief
Join
6
0
0
0.75 × 6 + 0.25 × 0 = 4.5
Safe
4
4
4
0.75 × 4 + 0.25 × 4 = 4
Joint project has no dominant action. Safe has the better minimum; Join wins under the displayed seventy-five-percent belief.
Strict dominance is stronger because every possible opposing action gives a positive gain. If Row assigns seventy-five percent to Column maintaining in the maintenance game, Maintain averages two and a quarter. Skip averages four, from three quarters of five plus one quarter of one. The gain is one and three quarters. At a zero-percent maintenance belief the gain remains one, and at one hundred percent it is two. Intermediate averages cannot reverse the sign when every component gain is positive. This argument needs probabilities for uncertainty, but does not need a particular forecast of the opponent.
Treat ties explicitly
An action weakly dominates another if it is at least as good against every opponent action and strictly better against at least one. Strict dominance requires a strict improvement in every comparison. Identical payoff rows do not weakly dominate each other under this course's convention.
Choose Ties matter below. Row's Top gives the same 2 as Bottom against Left, but gives 1 instead of 0 against Right. Top is weakly, not strictly, dominant. Against a certain Left, Bottom is still a best response. Weakly dominated actions can therefore appear in Nash equilibria.
Locate exactly where a weak comparison ties
In Ties matter, Row's Top and Bottom both give two against Left. Against Right, Top gives one and Bottom gives zero. Thus Top never does worse and sometimes does better. Weak refers to allowed equality, not a small advantage. A gain of one against Right suffices regardless of its likelihood. Conversely, if Top and Bottom gave identical utilities in both columns, neither would weakly dominate the other under this course's definition. The test requires some strict improvement as well as no losses.
Column's comparisons need equal care. Against Top, Left gives Column two while Right gives one. Against Bottom, Left and Right both give zero. Left weakly dominates Right but does not strictly dominate it. Both players therefore have a weakly dominant action, Top for Row and Left for Column. At Top, Left their utilities are two and two. That profile is one mutual best response. However, the presence of weak dominance does not mean every other action is always a mistake. The equalities leave room for other best responses against particular opposing choices.
A weakly dominated action can tie at the selected opponent action
Row / Column
Left
Right
Top
2, 2
1, 1
Bottom
2, 0
0, 0
At Bottom, Left
Stay
Switch alone
Row
2
2
Column
0
0
At Bottom, Left neither player can improve alone. Top weakly dominates Bottom, but its advantage occurs against Right.
Consider Bottom, Left. Row receives two and switching to Top would still give two. Column receives zero and switching to Right would still give zero. Neither can strictly improve alone. This meets the Nash definition introduced here, that no individual can obtain a higher utility by changing only their own action. Row's Bottom is weakly dominated by Top, yet it belongs to this equilibrium. There is no contradiction. Weak dominance permits a tie exactly where Row expects Column to play. Strict dominance would forbid that tie and therefore would rule out this situation.
The remaining weak-game cells sharpen the distinction. At Top, Right, Row gets one and switching to Bottom gives zero, so Row is best responding. Column gets one and switching to Left gives two, so Column is not. At Bottom, Right, Column is indifferent between zero and zero, while Row can raise utility from zero to one by switching to Top. Every best-response check must therefore include ties for the player concerned, even when the full cell fails Nash. A mark next to one utility does not certify the other player's incentive.
A small probability of the strict-comparison column changes Row's expected incentive. If Column chooses Right with ten-percent probability, Top averages ninety percent of two plus ten percent of one, or one point nine. Bottom averages ninety percent of two plus ten percent of zero, or one point eight. Top gains one tenth. If Right has zero probability, both average two. Weak dominance ensures no expected loss, but a strictly positive expected gain requires positive probability on at least one comparison with a strict gain. That qualification explains why certainty about a tying action matters.
Row's expected gain equals the probability of Right
Exact scale for Ties matter. A certain Left gives zero gain; any positive Right probability makes Top strictly better.
Do not round a small gain into a tie before reasoning. At one-percent probability of Right, Top averages one point nine nine and Bottom one point nine eight. The difference is one hundredth, still positive. In empirical work, an uncertain estimate might not justify that precision. In this stipulated game, however, exact inputs define exact comparisons. Keep those situations separate. Numerical equality inside the teaching model is an assumption to preserve, while a near equality in measured preferences may be a reason to examine sensitivity rather than assert a decisive behavioral prediction.
Compare incentives and outcomes
Inspect one action change at a time
Shared maintenance. Each cell lists Row, then Column utility.
Row / Column
Maintain
Skip
Maintain
Row: 3Column: 3
Row: 0Column: 5 ✓
Skip
Row: 5 ✓Column: 0
Row: 1 ✓Column: 1 ✓Both best responses
✓ means that player cannot improve by switching alone while the other action stays fixed. Ties receive marks too. A border identifies the selected cell.
Selected payoffs: 3, 3. Nash equilibrium: No. Pareto efficient among pure cells: Yes.
Unilateral comparison
Stay
Switch alone
Row; Column holds Maintain
3
5
Column; Row holds Maintain
3
5
Player
Strictly dominant action
Weakly dominant action
Row
Skip
Skip
Column
Skip
Skip
Under our definition, strictly dominant actions also satisfy weak dominance.
A game preset replaces the payoffs and resets both actions to the first listed choice. Action buttons change only that player’s choice. Utilities are stipulated. Pareto efficiency below compares only the four pure cells, not lotteries or outside options.
Ties matter. Each cell lists Row, then Column utility.
Row / Column
Left
Right
Top
Row: 2 ✓Column: 2 ✓Both best responses
Row: 1 ✓Column: 1
Bottom
Row: 2 ✓Column: 0 ✓Both best responses
Row: 0Column: 0 ✓
✓ means that player cannot improve by switching alone while the other action stays fixed. Ties receive marks too. A border identifies the selected cell.
Opponent choice
Row Top
Row Bottom
Left
2
2
Right
1
0
A tie in one column prevents strict dominance.
Read the checks as separate player comparisons
Choose Ties matter in the matrix explorer. The preset resets Row to Top and Column to Left. Read the table of unilateral comparisons before moving either player. Row can stay at two or switch to two. Column can stay at two or switch to one. The selected profile therefore has no profitable unilateral deviation, even though Row's best response is not unique. Now choose Row Bottom, leaving Column Left. Row still has two versus two, while Column has zero versus zero. The Nash status remains affirmative because a tie never counts as a strict improvement.
Keep Row Bottom and choose Column Right. Both selected utilities are zero. The checks show that Column cannot gain by switching alone, but Row can gain one by choosing Top. Choose Row Top to inspect that deviation directly. Its new pair is one and one, not one and zero, because a change of cell can affect both players. Only Row's gain establishes Row's incentive. This matters when reading pairs quickly. A player evaluates the change in their own coordinate, even if the other coordinate happens to move in the same direction.
Ties receive best-response marks even when the cell fails Nash
Row / Column
Left
Right
Top
2, 2 · R and C
1, 1 · R only
Bottom
2, 0 · R and C
0, 0 · C only
R and C identify individual best responses in Ties matter. Only cells containing both letters pass the Nash test.
The dominance summary does not depend on which cell you select. In Ties matter it continues to name Top and Left as weakly dominant while you move between cells. That is because dominance compares complete action alternatives across every opponent action. The selected-cell status instead checks just the current profile. In Shared maintenance, the summary names Skip as both strictly and weakly dominant. This is intentional. Our weak definition requires nonnegative differences everywhere and a positive difference somewhere, so strictly positive differences everywhere satisfy it too. Strict and weak labels are not mutually exclusive categories.
Choose Joint project to see all dominance entries become None. The reset selects Join, Join, a mutual best response. Select Safe for both players and inspect a second mutual best response. Safe gives four against Safe while a unilateral Join gives zero. Neither equilibrium relies on a dominant action because each player's best response changes with the opponent's action. The selected cell can therefore pass Nash while the dominance summary remains empty. This is a failure of dominance to settle the choice, not evidence that the matrix lacks a solution or that the controls missed a comparison.
Choose Match or differ for a contrasting failure. Row wants matching actions, while Column wants different actions. From Left, Left, Column can gain two by switching to Right. From Left, Right, Row can gain two by switching to Right. From Right, Right, Column gains two by switching to Left, and from Right, Left, Row gains two by switching to Left. The resulting cycle is a diagram of unilateral incentives, not an animated prediction of repeated play. Every pure cell fails the mutual-best-response test, and neither player has a dominant pure action.
Matching and mismatching incentives leave no pure resting cell
Match or differ has a profitable deviation from every pure cell. Arrows describe incentives, not observed repeated behavior.
Use these presets to distinguish three conclusions that are easy to conflate. A particular action can be best against the current opponent action. It can also be best against all opponent actions, which is dominance. Finally, a cell can contain a best response for each player, which is Nash equilibrium. The check marks display the first comparisons, the summary aggregates them into dominance claims, and the cell status checks their overlap. The explorer does not delete actions, search larger games, or infer what people believe. Its limited controls make these logical relationships visible without adding those capabilities.
Deletion depends on reasoning about others
A strictly dominated action cannot maximize expected utility against any probability distribution on the displayed opponent actions. After removing such an action, another action may become dominated in the reduced game. Iterated deletion relies on assumptions about others' rationality and their beliefs about each other.
Common knowledge goes beyond everyone knowing a fact: everyone knows it, knows everyone knows it, and so on. This is an idealization, not established by calling a player rational. Weak-dominance deletion can discard equilibria and depend on deletion order. Do not use it as if it had every property of strict deletion.
Track what deletion removes and what it assumes
Deleting an action means removing it from a reduced game used for analysis, not physically preventing a real person from choosing it. A first strict-dominance deletion uses only that player's preferences over the displayed alternatives. Further deletions may require confidence that other players avoid their own dominated actions. To see the difference, consider a separate paper game. Row has Top and Bottom, Column has Left and Right. Its payoff pairs in matrix order are three and three, zero and one, two and zero, and one and two. This is not an additional explorer preset.
In that paper game, Row prefers Top against Left, three rather than two, but Bottom against Right, one rather than zero. Column prefers Left against Top, three rather than one, but Right against Bottom, two rather than zero. Neither player has a dominated action initially. A tempting argument says 'Column will choose Left because three is their largest payoff,' then deletes Right, then declares Top optimal. The first deletion has no dominance justification. Right is Column's strict best response to Bottom. A conclusion reached after an unsupported restriction does not become valid because the final reduced comparison is simple.
One payoff revision justifies a strict deletion that previously failed
Original paper game
Row / Column
Left
Right
Top
3, 3
0, 1
Bottom
2, 0
1, 2
Revised paper game
Row / Column
Left
Right
Top
3, 3
0, 1
Bottom
2, 3
1, 2
Paper games, not presets. Only Column's Bottom-Left utility changes. Left then strictly beats Right under either Row action.
For a genuine strict-deletion example, change only Column's payoff at Bottom, Left from zero to three. Left now beats Right for Column against Top by two and against Bottom by one. Delete Right on that basis. In the remaining Left column, Row prefers Top by one. This second comparison uses Row's expectation that Column will not choose the strictly dominated Right. Row's own rationality alone did not supply that belief. The resulting Top, Left profile passes both original unilateral checks, but the route to it used an assumption about the other player's reasoning.
Weak deletion has a different risk even without long chains. Return to the actual Ties matter numbers on paper. Deleting Row Bottom first removes the Bottom, Left equilibrium already checked. The remaining Top row lets Column prefer Left strictly. If instead you delete Column Right first, the remaining Left column gives Row two under either action. Both Top and Bottom remain best responses in that reduced game. Even if further deletions can reach a common smaller outcome under some conventions, the surviving action sets at this stage differ and one deletion has already discarded an original equilibrium.
This example also shows why deleting a weakly dominated action can create a newly strict comparison. Column's Left beats Right only weakly in the full game because Bottom gives a zero-zero tie. Removing Bottom leaves only the strict two-versus-one comparison at Top. That new strictness is a property of the reduced game, not evidence that Right was originally strictly dominated. Always record which rows and columns a claim uses. The explorer's dominance table refers to the full current preset. It does not recompute a sequence of reductions or certify arbitrary deletion orders.
Weak deletion changes which comparisons remain
Delete Bottom first
Remaining row
Left
Right
Top
2, 2
1, 1
Column comparison
2 is better
1 is worse
Delete Right first
Remaining row
Left
Top
2, 2
Bottom
2, 0
Row comparison
2 equals 2
These are paper reductions of Ties matter. Removing Bottom discards an original equilibrium; removing Right first preserves Row's tie.
Common knowledge of rationality is stronger than each player being rational. It includes each knowing the others are rational, knowing that they know this, and continuing through further levels of knowledge. Iterated arguments can depend on such beliefs because each step uses conclusions about what others will rule out. A real player may understand their own incentives while doubting someone else's information or reasoning. Report the surviving incentive comparisons and the beliefs needed to justify additional restrictions, especially when ties permit several responses.
Exercises
Test a conditional best response
In Joint project, is Join strictly dominated by Safe? Test both opponent choices.
Hint
Compare 6 with 4, then 0 with 4.
Worked answer
Joint project. Each cell lists Row, then Column utility.
Row / Column
Join
Safe
Join
Row: 6 ✓Column: 6 ✓Both best responses
Row: 0Column: 4
Safe
Row: 4Column: 0
Row: 4 ✓Column: 4 ✓Both best responses
✓ means that player cannot improve by switching alone while the other action stays fixed. Ties receive marks too. A border identifies the selected cell.
Column chooses
Better row action
Comparison
Join
Join
6 > 4
Safe
Safe
4 > 0
Different columns favor different actions.
No. Join is better against Join, 6 instead of 4, but Safe is better against Safe, 4 instead of 0. Neither action dominates the other. Predicting a choice requires more than checking a single column.
Can a weakly dominated action survive?
In Ties matter, check Bottom, Left. Can either player strictly improve alone? What happens if you delete Bottom first?
Hint
Ties are best responses even when another action is equally good.
Worked answer
Ties matter. Each cell lists Row, then Column utility.
Row / Column
Left
Right
Top
Row: 2 ✓Column: 2 ✓Both best responses
Row: 1 ✓Column: 1
Bottom
Row: 2 ✓Column: 0 ✓Both best responses
Row: 0Column: 0 ✓
✓ means that player cannot improve by switching alone while the other action stays fixed. Ties receive marks too. A border identifies the selected cell.
Player
Stay
Switch alone
Row
2
2
Column
0
0
Weak deletion can remove a mutual best response.
Row gets 2 and would still get 2 after moving to Top. Column gets 0 and would still get 0 after moving to Right. Neither strictly improves, so this is a Nash equilibrium. Deleting the weakly dominated Bottom removes it. This does not contradict the exclusion of strictly dominated actions.
Sources and further reading
Selected reference passages support the definitions and algorithms below. The games, explanations, and implementations are original. Terminology and assumptions are stated within each lesson; source review is not an independent audit of this course. Optional links are not runtime dependencies.
Selected definitions of strict and weak dominance, and discussion of beliefs about other players. Weak dominance has different equilibrium-preservation properties from strict dominance.
Selected transcript passages on players, strategies, preferences, and how alternative payoffs change the game. Examples here are original, not observations of learner behavior.